Multiplier VS. mixer

-- and a mixer, because it isn't a plain multiplier, often outperforms a multiplier at the task of mixing RF signals, generally for a lower price.

This gets into semantics and convention. Historically, a multiplier has one input and 1 output. So, F(out)= nF(in) where n is an integer. It is not amplitude linear, but was widely used in FM transmitters. A mixer has two inputs, is generally linear, and the output contains the sum and difference frequencies of the two inputs. A mixer does not have to be linear; a logic Exclusive-OR gate is an excellent mixer. Any of these reqquires a filter at the output to select the correct signal.

Tam

An RF mixer is never linear. Non-linearity is a *requirement* for correct amplitude modulation. The only valid context for "linear" in RF mixer discussions is in discussions of the transmitter's signal envelope and the received signal waveform; they are essentially the same, so the result of the modulation is considered linear.

Don

"amoyou" a écrit dans le message de news: snipped-for-privacy@z14g2000cwz.googlegroups.com...

A multiplier calculates A(t)xB(t), A and B being analog signals

A mixer is a device able to upconvert or downconvert a RF signal, thanks to a multiplication. But usual (diode-based) mixers calculate something like A(t) x sign(B(t)), and not A(t)xB(t).

So a multiplier IS a mixer, but a mixer is often not an actual plain multiplier.

Cheers, Robert

OK, I used the term loosly, meaning the amplitude and sideband information are retained.

Tam

I read in sci.electronics.design that Tam/WB2TT wrote (in ) about 'Multiplier VS. mixer', on Wed, 21 Sep 2005:

It's also a four-quadrant multiplier.

I read in sci.electronics.design that Tam/WB2TT wrote (in ) about 'Multiplier VS. mixer', on Wed, 21 Sep 2005:

I don't know that it's 'loose'. An analogue mixer (frequency-changer) is internally non-linear but externally linear. You can tell, because people want to know the overall intermodulation distortion characteristic, although they call it 'third-order intercept'.

OTOH, an audio mixer had better be linear, linear, LINEAR. Unless it has fuzz (and/or several other toys) built-in as a feature. That sort of mixer is an adder, not a multiplier.

to the question #1:

mixing occurs whenever you apply a signal (like a simple ideal sinusoidal voltage) to the input of a non-linear block, meaning a black-box with a non-linear descriptive function, or a non-linear relationship between output and input. (Note that you cannot say a non-linear transfer function, because the concept of a transfer function is based on the hypothesis of linearity)

Any non-linearity is ok for mixing, but where does multiplication come out? Consider a signal as the collection of its frequency components, i.e. spectrum lines, which you can represent with a sinusoidal signal. Well, good old mathematicians assure us that any non-linearity (among those physically found in any actual device) can be expanded in a power serie like:

Fn(x) = x^0 + x^1 + x^2 +....+x^n

eventually infinite. And here are the multiplications: x^2 = x * x , and so on.. So, the multiplications are required for mixing and a multiplier is a mixer.

to the question #2:

Suppose to take a multiplier and to apply the two frequencies you want to mix f1 and f2 to its inputs. If the multiplier is ideal, doing the math shows that

A1*cos(2pi*f1 * t+phi1) * A2*cos(2pi*f2 * t+phi2)

gives exactly what you want: f1+f2 and f1-f2, then with a simple filter you keep the one you like. unfortunately multipliers are not ideal because the single devices are themself non-linear (besides the whole circuit) and you'll find that the output signal will have different terms (with suitable multiplicative constants)

K , x1,x2, x1^2,x2^2,x1*x2,x1^2 * x2 , x1*x2^2 .... | \\ / \\ / \\ / | \\/ \\ / \\ / order0 order1 order 2 order 3

with typically at lease three or four significant orders. if x1 and x2 are sinusoids, yuo'll have: f1 +/- f2, 2*f1 +/- f2, f1 +/- 2*f2, and much much more All of this is unacceptable if you have to do frequency conversione in a transceiver, because you'll spread "dirt" frequencies all around the spectrum !! ==> no FCC certification ==> no market ==> no money. Moreover it would also be extremely expensive to buil a filter capable of filtering out the extra frequency you get.

So, what's a mixer? it's a clever circuit which performs x1 * x2 in a particular way, so that the multiplicative constants of the higher order terms are __zero__ and therefore do not contribute. To see how this is possible, look for: "balanced mixers" --> zeros out unwanted order 1 terms "double-balanced mixers" --> zeros out unwanted order 1 and 2 terms "triple-balanced mixers" --> guess what?

Have fun,

Roberto

-------------------------------- There are only 10 types of people in the world: those who understand binary and those who don't.

Don (and Phil)

As John more or less stated: With an LO attached, at the terminals a good RF mixer is linear from the RF port to the IF port. No quotes needed -- if you model it as ideal and subject it to the test for a linear system it passes, and if the data sheet doesn't express the deviations from ideal (compression and 3rd-order intermod levels) then you don't trust the part.

Of course if you do a complete model it isn't linear, but then even a perfect vacuum isn't nonlinear if the field strengths get high enough.

The mathematical propellorhead term for a good multiplier is "bilinear", i.e. the output depends linearly on each of two inputs. It isn't true that the internals have to be nonlinear--e.g. a Gilbert cell with linearizing diodes, where the multiplication comes from the very accurately bilinear current splitting of the diff pairs. Even if you leave off the diodes, a diff pair is linear in one input (the emitter current) and has a tanh characteristic in the other (delta-Vbe).

Cheers,

Phil Hobbs

r(t) * (some periodic function of time) is a linear operation. Check it against the definition of a linear system and see.

r(t) * (some periodic function of time) is _exactly_ what a mixer is supposed to do.

Perhaps you cannot _implement_ a mixer without internal nonlinearities, but you can certainly _describe_ a mixer as a linear device.

You cannot describe a mixer as a linear device, but you can describe the result of modulation via the non-linear mixer, as linear.

In other words, the linear *result* is a function of the mixer's non-linearity.

Don

You can't get new frequencies *with a single input port* without there being nonlinearity, but that doesn't exhaust the possibilities. Consider an ideal multiplier (which I assume we both agree can generate sum and difference frequencies), producing Vo=k*V1*V2.

Superposition:

Vo(f+g, V2) = k(f+g)*V2 (port 1 obeys superposition) Vo(V1, f+g) = k*V1*(f+g) (port 2 obeys superposition)

Scale invariance:

Vo(A*f, V2) = k(A*f)*V2 (port 1 is scale-invariant) Vo(V1, A*f) = k*V1*(A*f) (port 2 is scale-invariant)

Since it obeys scale invariance and superposition, an ideal multiplier is linear in V1 and linear in V2, hence it's called "bilinear". Lots of physical effects have this property too, e.g. the Pockels electrooptic effect.

Cheers,

Phil Hobbs

I read in sci.electronics.design that Phil Hobbs wrote (in ) about 'Multiplier VS. mixer', on Thu, 22 Sep 2005:

You cannot get new frequencies without there being non-linearity.

Yea, so?

Do you know how a linear system is defined? It's a precise one, and easy to prove/disprove if you can get the math to behave. Do you know what superposition is, and how it relates to the definition of a linear system?

I'm not arguing that a multiplier _in general_ is a linear element -- it ain't. I'm just saying that a mixer _as used in a radio_ is best treated as a linear time-varying element, not a nonlinear element.

Huh? Draw me graph of the transfer function of two variables multiplied together. You may use three dimensions, if need be. At least along some plane, it looks rather quadratic to me!

Yes

I believe I do.

Given the very clear definition for a linear system, RF mixing is not linear.

"A system is linear if its response is directly proportional to excitation, for every part of the system."

The non-linear component(s) necessary for RF mixing prevents meeting the requirements of that definition.

Why are you trying to force-fit a non-linear process into the mold of a linear process?

Don

That's a FREQUENCY multiplier. The multiplier being discussed elsewhere in this thread is a two-input device whose output voltage is proportional to the product of the voltages at its two inputs. The classic IC example of this is (was?) the Motorola/On Semi MC1496. I wanted to call this a "voltage multiplier" but that's a common name for a type of power supply circuit.

There's the mixer in radio receivers, in between the antenna or RF amplifier and the IF stage(s), and there's the completely different device for setting levels of audio signals also called a mixer.

So now in this thread we have two different things called mixers, and three different things called multipliers. What else can we add (or mix, or multiply) to make it more confusing???

First, I'm talking about the input - output relationship only, not what's going on inside. I've pointed that out multiple times in case that's where the disagreement is.

Second, your definition is for a linear _time-invariant_ system. The definition of just linearity is for a system h(x, t) that turns a signal x(t) into a signal y(t) the system is linear if

h(x1(t) + x2(t), t) = h(x1(t), t) + h(x2(t), t).

I.e. the system satisfies superposition. If you can find a serious, professional-level book on signal processing that contradicts this let me know, we can go from there. With the output taken at the LO port and the input put into the RF port a mixer satisfies this condition and is therefore linear.

A time_invariant system is one that satisfies

h(x(t), t + t0) = h(x(t), t).

A mixer doesn't satisfy this (thankfully, or it wouldn't do its job), so it's time-varying.